Is it possible to make a mesh symmetrical without mirror modifier?
Yes.

It is possible to make a mesh symmetrical without a Mirror Modifier.

Simply set your Pivot Point to be your 3D Cursor, place the 3D Cursor at the center of symmetry, duplicate the geometry to be mirrored (ShiftD), and scale (S) -1 on the axis of symmetry (in this case the X axis X).

You can set the Pivot Point to the 3D Cursor here:



Take this example of half a monkey:



Finally, pressing W and choosing Remove Doubles can get rid of overlapping vertices and weld the mesh halves together.

You'll want to Recalculate Normals after doing this (CtrlN) because the face normals get flipped inside-out in the scaling process.